## Physics 7C: Electromagnetic waves, physical optics, relativity and quantum physics Pole and Barn

Note: the pictures look a lot better if you look at them separately (view the image alone).

The pole and barn "paradox" can be stated as follows. A pole and a barn are taken to have the same length when both are at rest. Someone (runner) then takes the pole and runs towards the barn with it. Someone else (watcher) stands by the barn and watches what happens.

What do they expect?

• The runner says: I'm running towards the barn, so it has a velocity relative to me and will be Lorentz contracted. So it will be smaller than the pole in length and the pole won't fit inside. I hope the far door is open!!
• The watcher says: What's the problem? The person with the pole is moving relative to me with some velocity v, and so the pole will be shortened, it will fit in easily!

As usual with relativity they are both right. The point is that when you measure the length of something, you measure the distance between its ends at some fixed time. That is, you want the distance between two positions at the same time, according to you. Someone moving relative to you will have a different idea of "fixed time".

Consider the spacetime diagram below.
The black lines are the frame of the watcher, and the two red vertical lines are the positions of the end of the barn as time changes. The ends are staying put in this frame so the ends of the barn trace out vertical lines, i.e. with the same position x for each of them at any time. The two black circles indicate the two events:

• front of pole leaves barn
• back of pole enters barn
and the pole is said to "fit" if the back of the pole gets in the barn before the front leaves it.
The two parallel tilted lines are the ends of the pole, both moving with the same velocity, and separated by a distance equal to the length of the pole in the rest frame of the pole. The length scales are different along the x axis and x' axis (the same distance along the x and x' axis will look longer on the x' axis).

Now, consider where the ends of the pole are at fixed time t, in the frame of the watcher. To find out what happens When the back end of the pole enters the barn, the front of the pole has not yet left, the pole "fits".

Consider the frame of the watcher. Constant time for the watcher, when the back of the pole enters the barn, is shown by a black line. At this time the front of the pole has already left the barn, so the pole doesn't fit.

The reason there is an apparent paradox is that in the notion of fitting the pole into the barn is an implicit assumption that both people can agree what it means to measure the positions of the ends at the same time. They can't--two people who have a relative velocity will not agree on what is simultaneous in special relativity!

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